Question

For f(x) = {2x+3, x ≤ 1; -(x-2)²+1, x>1}; b) State all the x values at which the function is determined in part (b) by determining the domain of the function.

Answer 1

The domain of the piecewise function f(x) is all real numbers, as it is defined for x ≤ 1 and for x > 1, covering all x-values.

Step-by-step explanation:

The student's question asks for the domain of the piecewise function f(x) = {2x+3 for x ≤ 1; -(x-2)²+1 for x>1}. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the first part of the function, 2x+3, it is defined for all x-values less than or equal to 1. The second part of the function, -(x-2)²+1, is defined for all x-values greater than 1. Therefore, the domain of the entire function includes all real numbers x such that x is less than or equal to 1 and also those greater than 1. Since all real numbers meet this condition, the domain of f(x) is all real numbers, or in mathematical notation, (-∞, ∞).